ar X iv : m at h / 03 04 17 3 v 2 [ m at h . R T ] 2 5 Fe b 20 04 Quantum Groups , the loop Grassmannian , and the Springer resolution

We establish equivalences of the following three triangulated categories: Dquantum(g) ←→ D G coherent(Ñ ) ←→ Dperverse(Gr). Here, Dquantum(g) is the derived category of the principal block of finite dimensional representations of the quantized enveloping algebra (at an odd root of unity) of a complex semisimple Lie algebra g; the category D coherent(Ñ ) is defined in terms of coherent sheaves on the cotangent bundle on the (finite dimensional) flag manifold for G (= semisimple group with Lie algebra g), and the category Dperverse(Gr) is the derived category of perverse sheaves on the Grassmannian Gr associated with the loop group LG, where G is the Langlands dual group, smooth along the Schubert stratification. The equivalence between Dquantum(g) and D coherent(Ñ ) is an ‘enhancement’ of the known expression (due to Ginzburg-Kumar) for quantum group cohomology in terms of nilpotent variety. The equivalence between Dperverse(Gr) and D coherent(Ñ ) can be viewed as a ‘categorification’ of the isomorphism between two completely different geometric realizations of the (fundamental polynomial representation of the) affine Hecke algebra that has played a key role in the proof of the DeligneLanglands-Lusztig conjecture. One realization is in terms of locally constant functions on the flag manifold of a p-adic reductive group, while the other is in terms of equivariant K-theory of a complex (Steinberg) variety for the dual group. The composite of the two equivalences above yields an equivalence between abelian categories of quantum group representations and perverse sheaves. A similar equivalence at an even root of unity can be deduced, following Lusztig program, from earlier deep results of Kazhdan-Lusztig and KashiwaraTanisaki. Our approach is independent of these results and is totally different (it does not rely on representation theory of Kac-Moody algebras). It also gives way to proving Humphreys’ conjectures on tilting Uq(g)-modules, as will be explained in a separate paper.